A033997 -id:A033997 - cp's OEIS frontend

A061890 Squares that are the sum of initial primes.

100, 25633969, 212372329, 292341604, 3672424151449, 219704732167875184222756

Offset: 1

David W. Wilson, May 12 2001

The set of squares in A007504. - Zak Seidov, Oct 07 2015

			100 = 10^2 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23, so 100 is in the sequence.

G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
Carlos Rivera, Problem 9: Sum of first k primes is perfect square, The Prime Puzzles & Problems Connection.

Cf. A007504, A033997, A033998, A061888.

Cf. also A066527, A364691, A366269.

N:= 10^7: # to use primes up to N
P:= select(isprime, [2,seq(2*i+1, i=1..floor(N/2))]):
S:= ListTools:-PartialSums(P):
select(issqr,S); # Robert Israel, Feb 16 2015

s[n_] := Sum[Prime[i], {i, 1, n}];t := Table[s[n], {n, 20000}];Select[t, IntegerQ[Sqrt[#]] &] (* Carlos Eduardo Olivieri, Feb 24 2015 *)

lista() = {s = 0; forprime(p=2, ,s += p; if (issquare(s), print1(s, ", ")););} \\ Michel Marcus, Mar 10 2015

a(n) = A061888(n)^2.

Intersection of A000290 and A007504. - Zak Seidov, Oct 08 2015

Corrected by Vladeta Jovovic, May 21 2001
a(6) from Giovanni Resta, May 27 2003
Edited by Ray Chandler, Mar 20 2007

A364696 Nonnegative integers k such that the sum of the first k primes is a pentagonal number.

Original entry on oeis.org

0, 2, 77, 24587, 48070640, 471412484, 7471587112

Offset: 1

Paolo Xausa, Aug 03 2023

			2 is a term because the sum of the first 2 primes (2 + 3 = 5) is a pentagonal number.

Wikipedia, Pentagonal number.
Index to sequences related to polygonal numbers

Cf. A000326, A007504, A033997, A175133, A364691, A364695, A366270.

A364696list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[24(p+=Prime[k])+1]+1)/6],k,Nothing],{k,kmax}]]];A364696list[25000] (* Paolo Xausa, Oct 06 2023 *)

a(5) from Michel Marcus, Aug 04 2023

a(6)-a(7) from Hugo Pfoertner, Aug 04 2023

A033998 Numbers n such that the sum of the primes <= n is a square.

Original entry on oeis.org

23, 22073, 67187, 79427, 10729219, 3531577135439

Offset: 1

Calculated by Jud McCranie

nonn

			2+3+5+7+11+13+17+19+23 = 10^2 is a square, so 23 is in the sequence.

G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 3531577135439
C. Riveras, PrimePuzzles.Net, Problem 9: Sum of first k primes is perfect square

Cf. A000040, A033997, A061888, A061890.

Prime[#]&/@Flatten[Position[Accumulate[Prime[Range[710000]]],?(IntegerQ[ Sqrt[#]]&)]] (* This program will generate the first 5 terms of the sequence. *) (* _Harvey P. Dale, Jun 22 2013 *)

s=0;forprime(p=2,1e6,if(issquare(s+=p),print1(p", "))) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = Prime(A033997(n))

3531577135439 from Giovanni Resta, May 27 2003

Edited by Ray Chandler, Mar 20 2007

A175133 Sum of first a(n) consecutive primes gives a triangular number.

Original entry on oeis.org

3, 5, 217, 1065, 93448, 39545957, 240439822, 1894541497, 132563927578, 309101198255

Offset: 1

Ctibor O. Zizka, Feb 20 2010

A007504(a(n)) = A000217(j) for some j.

Numbers k such that Sum_{i=1..k} prime(i) = j*(j+1)/2, where prime(i) is i-th prime, and j an integer.

			k=3 is a term: 2+3+5=10, and 10=4*5/2 is a triangular number, j=4.
k=5 is a term: 2+3+5+7+11=28, and 28=7*8/2 is a triangular number, j=7.
k=217 is a term: 2+3+...+1327=133386, and 133386=516*517/2 is a triangular number, j=516.

Cf. A000040, A000217, A007504, A066527.

Cf. also A033997, A364696, A366270.

isok(n) = ispolygonal(sum(k=1, n, prime(k)), 3); \\ Michel Marcus, Oct 13 2018

a(6)-a(10) from Nathaniel Johnston, May 10 2011 (a(7)-a(10) based on comments by Donovan Johnson)

Name, comment and example clarified by Ilya Gutkovskiy, Aug 07 2023

A364695 Positive integers k such that the sum of the first k primes is a polygonal number of order greater than 2 (A090466).

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 53, 54, 56, 57, 62, 68, 70, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 97, 99, 103, 105, 106

Offset: 1

Paolo Xausa, Aug 03 2023

nonn

			5 is a term because the sum of the first 5 primes (2 + 3 + 5 + 7 = 28) is a triangular number.
7 is a term because the sum of the first 7 primes (2 + 3 + 5 + 7 + 11 + 13 = 58) is an octagonal number.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A007504, A033997, A090466, A175133, A364693, A364694, A364696.

A364693Q[n_]:=With[{d=Divisors[2n]},Catch[For[i=3,iJianing Song in A090466 *)
A364695list[kmax_]:=Flatten[Position[Map[A364693Q,Accumulate[Prime[Range[kmax]]]],True]];A364695list[100]

isok(k) = my(s = sum(i=1, k, prime(i))); for (j=3, s-1, if (ispolygonal(s, j), return(1))); \\ Michel Marcus, Aug 03 2023

A366270 Nonnegative integers k such that the sum of the first k primes is a hexagonal number.

Original entry on oeis.org

0, 5, 93448, 39545957, 240439822, 1894541497, 132563927578

Offset: 1

Paolo Xausa, Oct 06 2023

			5 is a term because the sum of the first five primes (2 + 3 + 5 + 7 + 11 = 28) is a hexagonal number.

Wikipedia, Hexagonal number.
Index to sequences related to polygonal numbers

Subsequence of A175133.

Cf. A000384, A007504, A033997, A364695, A364696, A366269 (corresponding hexagonal numbers).

A366270list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[8(p+=Prime[k])+1]+1)/4],k,Nothing],{k,kmax}]]];A366270list[10^5]

A061888 Numbers k such that k^2 is the sum of the first m primes for some m.

Original entry on oeis.org

10, 5063, 14573, 17098, 1916357, 468726713734

Offset: 1

David W. Wilson, May 12 2001

			10^2 = 2+3+5+7+11+13+17+19+23, so 10 is in the sequence.

G. L. Honaker Jr. and C. Caldwell, Prime Curios!: 9
Carlos Rivera, Problem 9: Sum of first k primes is perfect square, The Prime Puzzles and Problems Connection.

Cf. A033997, A033998, A061890.

a(n)^2 = A061890(n).

a(6) from Giovanni Resta, May 27 2003

Edited by Ray Chandler, Mar 20 2007

A264858 Integers k such that A007504(k) + 1 is a square.

Original entry on oeis.org

0, 17, 539, 652, 6420, 350857847

Offset: 1

Altug Alkan, Nov 26 2015

Integers k such that the sum of the first k primes + 1 is a square.
Integers k such that A014284(k+1) is a square.
In A110996, it is commented that a(6) > 250000, if it exists.
a(6) > 50000000, if it exists. - Jon E. Schoenfield, Nov 29 2015

			a(2) = 17 because A007504(17) + 1 = 440 + 1 = 441 is a square.

Cf. A007504, A014284, A033997, A089228, A110996.

t = Accumulate[Prime@ Range@ 100000]; Prepend[Select[Range@ 7000, IntegerQ@ Sqrt[t[[#]] + 1] &], 0] (* Michael De Vlieger, Nov 29 2015, after Zak Seidov at A007504 *)
Join[{0},Position[Accumulate[Prime[Range[6500]]]+1,?(IntegerQ[Sqrt[ #]]&)]] // Flatten (* _Harvey P. Dale, Feb 12 2018 *)

lista(nn) = { print1(0, ", "); s = 1; for(k=1, nn, s += prime(k); if(issquare(s), print1(k, ", ")););}

a(6) from Jinyuan Wang, Aug 09 2023

A274120 Squares that are the sum of the first k odd primes for some k.

Original entry on oeis.org

961, 1369, 1849, 4225, 263169, 130919364, 758451600, 29682949232484409

Offset: 1

K. D. Bajpai, Jun 10 2016

Intersection of A000290 and A071148. [Felix Fröhlich, Jun 10 2016]

a(9), if it exists, is larger than 2*10^26 and would require adding more than 3.6*10^12 primes. - Giovanni Resta, Jun 12 2016

			961 is in the sequence because 961 = 31^2. Also, 3+5+7+...+83+89 = 961.

G. L. Honaker Jr. and C. Caldwell, Prime Curios! : 961

Cf. A007504, A033997, A061890, A065091, A071148.

Select[s=0;Table[s=s+Prime[k],{k,2,15000000}],IntegerQ[Sqrt[#]]&]
Select[Accumulate[Prime[Range[2,100000]]],IntegerQ[Sqrt[# ]]&] (* Harvey P. Dale, May 26 2023 *)

my(s = 0); forprime(p=3, 1e10, s += p; if (issquare(s), print1(s, ", ")))

Second comment rephrased by Harvey P. Dale, May 26 2023

A353024 Largest k such that A007504(k) <= n^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47

Offset: 1

Joelle H. Kassir, Apr 17 2022

nonn

Cf. A000290, A007504, A337769, A350174, A033997.

first(N)=my(v=vector(N),s,k,n=1,n2=1); forprime(p=2,, s+=p; k++; while(s>n2, v[n]=k-1; if(n++>N, return(v)); n2=n^2)) \\ Charles R Greathouse IV, Apr 18 2022

a(n)=my(n2=n^2,s,k); forprime(p=2,, s+=p; k++; if(s>n2, return(k-1))) \\ Charles R Greathouse IV, Apr 18 2022

from sympy import prime
def a(n):
    k = 1
    total = 0
    while True:
        total += prime(k)
        if total > n**2:
            break
        k += 1
    return k-1

a(n) = A337769(n) - 1.
a(n) ~ sqrt(2)*n/sqrt(log n). - Charles R Greathouse IV, Apr 18 2022
a(n) = A350174(n^2). - Kevin Ryde, Apr 19 2022

cp's OEIS Frontend

A061890 Squares that are the sum of initial primes.

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A364696 Nonnegative integers k such that the sum of the first k primes is a pentagonal number.

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A033998 Numbers n such that the sum of the primes <= n is a square.

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A175133 Sum of first a(n) consecutive primes gives a triangular number.

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A364695 Positive integers k such that the sum of the first k primes is a polygonal number of order greater than 2 (A090466).

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A366270 Nonnegative integers k such that the sum of the first k primes is a hexagonal number.

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A061888 Numbers k such that k^2 is the sum of the first m primes for some m.

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A264858 Integers k such that A007504(k) + 1 is a square.

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